\(\bullet \)  The equidistribution assumption is satisfied for all totally ergodic systems if and only if (REF ) holds for all \(t_1,\ldots ,t_\ell \in \operatorname{Spec}(T)\) 
that are irrational or 0, but not all of them 0. In particular,
this is satisfied by collections of linearly independent polynomials with integer coefficients and zero constant terms (which are also known to be good for seminorm estimates), thus providing a simpler proof of the main result in [1]}. See also Theorem REF  below for an application to a vastly more general collection of Hardy field sequences.
